We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-Kähler manifold of $\mathrm {K3}^{[2]}$-type whose deformation space is smooth of dimension 10. Its moduli space is birational to a hyper-Kähler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-Kähler could lead to new examples of hyper-Kähler manifolds.

In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $-stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$, where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$, if $g(C)\leq 1$; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$.

We show that the derived category of a curve is embedded into the derived category of the moduli space of vector bundles on the curve of coprime rank and degree. We also generalize the semiorthogonal decomposition constructed by Narasimhan and Belmans-Mukhopadhyay. Finally, we produce a one-dimensional family of ACM bundles over the moduli space.

We provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers, we call it a positive Ulrich sheaf if this bilinear form is symmetric or Hermitian and positive-definite. In that case, our result provides a common theoretical framework for several results in real algebraic geometry concerning the existence of algebraic certificates for certain geometric properties. For instance, it implies Hilbert’s theorem on nonnegative ternary quartics, via the geometry of del Pezzo surfaces, and the solution of the Lax conjecture on plane hyperbolic curves due to Helton and Vinnikov.

The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$.

We introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira and Moreira, Oblomkov, Okounkov and Pandharipande. We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

In this paper, we prove a stronger form of the Bogomolov–Gieseker (BG) inequality for stable sheaves on two classes of Calabi–Yau threefolds, namely, weighted hypersurfaces inside the weighted projective spaces $\mathbb {P}(1, 1, 1, 1, 2)$ and $\mathbb {P}(1, 1, 1, 1, 4)$ . Using the stronger BG inequality as a main technical tool, we construct open subsets in the spaces of Bridgeland stability conditions on these Calabi–Yau threefolds.

Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of Brill–Noether loci to construct examples of Brill–Noether loci on fibered surfaces. Results concerning the emptiness of Brill–Noether loci follow as a consequence of a generalization of Clifford’s Theorem for rank two bundles on surfaces.

We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

We study the moduli space of rank 2 instanton sheaves on ℙ3 in terms of representations of a quiver consisting of three vertices and four arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter θ for which the corresponding quiver representation is θ-stable (in the sense of King), and that the space of stability parameters has a non-trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stability parameters with respect to which every representation corresponding to a rank 2 instanton sheaf of charge 2 is stable and provide a complete description of the wall-and-chamber decomposition for representation corresponding to a rank 2 instanton sheaf of charge 1.

We express nested Hilbert schemes of points and curves on a smooth projective surface as ‘virtual resolutions’ of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa–Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom–Porteous-like Chern class formulae.

We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

Let X be a non-singular irreducible complex projective curve of genus g ≥ 2. The concept of stability of coherent systems over X depends on a positive real parameter α, given then a (finite) family of moduli spaces of coherent systems. We use (t, ℓ)-stability to prove the existence of coherent systems over X that are α-stable for all allowed α > 0.

In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.

For a pair $(R,I)$, where $R$ is a standard graded domain of dimension $d$ over an algebraically closed field of characteristic 0, and $I$ is a graded ideal of finite colength, we prove that the existence of $\lim _{p\rightarrow \infty }e_{HK}(R_{p},I_{p})$ is equivalent, for any fixed $m\geqslant d-1$, to the existence of $\lim _{p\rightarrow \infty }\ell (R_{p}/I_{p}^{[p^{m}]})/p^{md}$. This we get as a consequence of Theorem 1.1: as $p\longrightarrow \infty$, the convergence of the Hilbert–Kunz (HK) density function $f(R_{p},I_{p})$ is equivalent to the convergence of the truncated HK density functions $f_{m}(R_{p},I_{p})$ (in $L^{\infty }$ norm) of the mod $p$reductions$(R_{p},I_{p})$, for any fixed $m\geqslant d-1$. In particular, to define the HK density function $f_{R,I}^{\infty }$ in char 0, it is enough to prove the existence of $\lim _{p\rightarrow \infty }f_{m}(R_{p},I_{p})$, for any fixed $m\geqslant d-1$. This allows us to prove the existence of $e_{HK}^{\infty }(R,I)$ in many new cases, for example, when Proj R is a Segre product of curves.

Let $\unicode[STIX]{x1D709}$ be a stable Chern character on $\mathbb{P}^{1}\times \mathbb{P}^{1}$, and let $M(\unicode[STIX]{x1D709})$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ with Chern character $\unicode[STIX]{x1D709}$. In this paper, we provide an approach to computing the effective cone of $M(\unicode[STIX]{x1D709})$. We find Brill–Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\unicode[STIX]{x1D709})$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^{1}\times \mathbb{P}^{1}$.

Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{\text{ort}}^{0}\left( r,\,n \right)$ of stable rank $r$ orthogonal vector bundles on ${{\mathbb{P}}^{2}}$, with Chern classes $\left( {{c}_{1}},\,{{c}_{2}} \right)\,=\,\left( 0,\,n \right)$ and trivial splitting on the general line, is smooth irreducible of dimension $\left( r-2 \right)n\,-\,\left( _{2}^{r} \right)$ for $r\,=\,n$ and $n\,\ge \,4$, and $r\,=\,n-1$ and $n\,\ge \,8$. We speculate that the result holds in greater generality.

We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalises the one on projective spaces. This is done by generalising the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterisation of these sheaves in the particular case of projective spaces.

Let (X, H) be a polarized smooth projective surface satisfying H1(Χ, OΧ) = 0, and let Ƒ be either a rank 1 torsion-free sheaf or a rank 2 μH-stable vector bundle on Χ. Assume that c1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheaf Ƒ[2] associated with Ƒ on the Hilbert square Χ[2] is μ-stable with respect to a certain polarization.